Method and system for latent demand modeling for a transportation system

ABSTRACT

A method and system for identifying demand in a transportation system. A boarding count model is determined based upon passenger arrival information, and a geographic and time-specific generalized boarding model is determined based upon the boarding count model as well as information related to a plurality of stops on a route in the transportation system. For each of the plurality of stops, an approximated uniform arrival model is determined based upon the generalized arrival model and a time period between arriving vehicles at a specific stop, an instantaneous demand model is determined based upon the uniform arrival model, and a probability of no demand model is determined based upon the uniform arrival model. A report including the instantaneous demand and the probability of no demand determined can be generated. Based upon the report, various operational parameters for the transportation system can be manually or automatically adjusted.

BACKGROUND

The present disclosure relates to modeling demand in a transportationsystem, such as a public bus, train or plane system. More specifically,the present disclosure relates to latent demand modeling as a functionof the time of the day and the day of the week for a transportationsystem.

Many service providers monitor and analyze analytics related to theservices they provide. One important analytic related to efficientoperation is travel demand for a transportation system or a particularroute in a transportation system. For example, public transportationvehicles may be equipped with an automated passenger counter configuredto measure passengers boarding or alighting a vehicle at a particularstop. However, data from automated passenger counters is not collectedregularly, and thus the information is difficult to accurately correlateto time and place. Additionally, if no one is at a stop, the vehicletypically will not stop unless there is a passenger wanting to get offthe vehicle. Thus, such stops may be ignored completely and there is noregistration of the stop with the automated passenger counter.

Additionally, public transportation vehicle routes are run irregularlythroughout the day, and some routes are not run at all at certain hourssuch as late at night or early in the morning, e.g. from 2:00 AM to 5:00AM. Thus, the actual number of passengers picked up at a stop, i.e., thedemand at that stop, is not only dependent upon the time of the day butalso the interval between vehicles servicing that stop. A longerinterval will result in a higher number of passengers. However, thisincrease in passengers may not be related to the population or overalldemand of the stop. Rather, the increase may be a result of a longertime interval between vehicles servicing that particular stop. As such,using existing technology and techniques to estimate demand resultsprovides an incomplete analysis when modeling demand as a function oftime of day and day of week.

SUMMARY

In one general respect, the embodiments disclose a method of identifyingdemand in a transportation system. The method includes determining aboarding count model based upon passenger arrival information anddetermining a geographic and time-specific generalized boarding modelbased upon the boarding count model as well as information related to aplurality of stops on a route in the transportation system. For each ofthe plurality of stops, the method includes determining an approximateduniform arrival model based upon the generalized arrival model and atime period between arriving vehicles at a specific stop, determining aninstantaneous demand model based upon the uniform arrival model, anddetermining a probability of no demand model based upon the uniformarrival model. The method also includes generating a report including atleast an indication of instantaneous demand determined based upon theinstantaneous demand model and an indication of probability of no demanddetermined based upon the no demand model, and presenting the report.

In another general respect, the embodiments disclose a system foridentifying demand in a transportation system. The system includes aprocessing device, a display device operably connected to the processingdevice, and a computer readable medium in communication with theprocessing device. The computer readable medium includes one or moreprogramming instructions for causing the processing device to determinea boarding count model based upon passenger arrival information anddetermine a geographic and time-specific generalized boarding modelbased upon the boarding count model as well as information related to aplurality of stops on a route in the transportation system. For each ofthe plurality of stops, the one or more instructions cause theprocessing device to determine an approximated uniform arrival modelbased upon the generalized arrival model and a time period betweenarriving vehicles at a specific stop, determine an instantaneous demandmodel based upon the uniform arrival model, and determine a probabilityof no demand model based upon the uniform arrival model. The one or moreinstructions further cause the processing device to generate a reportthat includes an indication of instantaneous demand determined basedupon the instantaneous demand model and an indication of probability ofno demand determined based upon the no demand model, and display thereport on the display device.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 depicts a sample flow chart for modeling demand for atransportation system according to an embodiment.

FIG. 2 depicts an example of a plot of observed and estimated passengersboarding for a particular transportation route according to anembodiment.

FIG. 3A depicts a level plot illustrating estimated demand at a stop ona transportation system as a function of hour and day of week accordingto an embodiment.

FIG. 3B depicts a level plot illustrating a probability of zero demandfor a stop on a transportation system according to an embodiment.

FIG. 4 depicts various embodiments of a computing device forimplementing the various methods and processes described herein.

DETAILED DESCRIPTION

This disclosure is not limited to the particular systems, devices andmethods described, as these may vary. The terminology used in thedescription is for the purpose of describing the particular versions orembodiments only, and is not intended to limit the scope.

As used in this document, the singular forms “a,” “an,” and “the”include plural references unless the context clearly dictates otherwise.Unless defined otherwise, all technical and scientific terms used hereinhave the same meanings as commonly understood by one of ordinary skillin the art. Nothing in this disclosure is to be construed as anadmission that the embodiments described in this disclosure are notentitled to antedate such disclosure by virtue of prior invention. Asused in this document, the term “comprising” means “including, but notlimited to.”

As used herein, a “computing device” or “processing device” refers to adevice that processes data in order to perform one or more functions. Acomputing device may include any processor-based device such as, forexample, a server, a personal computer, a personal digital assistant, aweb-enabled phone, a smart terminal, a dumb terminal and/or otherelectronic device capable of communicating in a networked environment. Acomputing device or processing device may interpret and executeinstructions. Unless specifically limited, reference to any device canrefer to a single device and/or a group of devices which work togetherto implement a process.

A “mathematical model,” or simply “model,” refers to a process ofdeveloping a mathematical representation of one more variables and oneor more relationships that exists between those variables. A variable,as used herein, refers to abstractions of quantifiable parameters ofinterest that are either known or are being solved for in the model. Arelationship, as used herein, refers to algebraic operators, functionsand algorithms and other similar mathematical operators.

The present disclosure is directed to a method and system for modelingdemand for a transportation system using Bayesian latent modelingtechniques. Based upon information collected and modeling for aparticular route or stop in the transportation system, instantaneousdemand for that stop or route can be modeled for a particular time ofday and day of week, as well as the probability of no demand occurringat any particular time of day and day of week. Using the techniques asdescribed herein, a public transportation company can monitor demand ateach stop along its provided routes to determine if, for example, stopsshould be eliminated, additional stops should be added, vehicle sizescan be adjusted, and other similar actions that can impact systemefficiency and reduce and/or maximize operating expenses.

FIG. 1 illustrates a flow chart showing a sample process for modelingdemand for a particular transportation system. For discussion purposesonly, a public transportation system including multiple buses operatingon varying routes, each route include multiple stops will be described.However, it should be noted that such a transportation system isdescribed by way of example only. The processes, systems and methods astaught herein may be applied to any environment where performance basedmetrics and information are collected for later analysis, and providedservices may be altered accordingly based upon the collected informationto improve demand-based efficiency.

As shown in FIG. 1, a computing device may determine 102 a currentboarding count model for a particular stop in the public transportationsystem. Passenger information can be obtained from an automaticpassenger counter device associated with each vehicle in thetransportation system. Inspection of typical passenger data for a publictransportation system may show fairly sparse demand at certain times ofthe day. For example, near a city, public transportation demand isgreatest at rush hours (e.g., 7:00-10:00 AM and 4:00-7:00 PM). At timesoutside of the rush hour ranges, demand can be reduced to near zero. Assuch, determining 102 a current boarding count model for a particularstop may be done using a zero inflated Poisson distribution, i.e., adistribution that provides for frequent zero-valued observations. Inthis case, the processing device may determine 102 the current boardingmodel using the following equation:f(x; p, λ)=pI(x=0)+(1−p)Pois(x,λ)  (1)where x is the number of passengers boarding the vehicle at a given timeinterval. Demand is thus determined as a rate of passengers boarding asa function of time. For example, the demand can be modeled as a Poissonarrival process that is conditioned on a latent variable of non-zerodemand.

However, in a typical public transportation system, different routesexhibit different demand rates as a function of geography. For example,stops located closer to higher population centers are expected toexhibit a higher demand than stops located in rural or sparselypopulated areas. To account for such variation, the processing devicemay generalize 104 the current boarding count model for a particularroute using the following equation:f(x;p,λ _(i))=pI(x=0)+(1−p)Pois(x,λ _(i)), i=1, . . . , n  (2)where n is the number of stops along a particular route. Thus, theprocessing device can generalize 104 the boarding count model to producea generalized boarding model accounting for each stop along a route or,more generally, each stop serviced by the transportation system, thusaccounting for the geographic impact on demand throughout thetransportation system.

Additionally, beyond merely accounting for geographic impact on demand,the time of the day, as well as the time between scheduled stops, canimpact demand. In particular, the longer the time interval between avehicle arriving at a particular stop, the larger the resulting demandwill be at the stop. In practice, arrival rates increase closer to thescheduled vehicle arrival time. However, for modeling, uniform arrivalover the time interval can serve as an accurate approximation. Thus, thecomputing device may determine 106 an approximated uniform arrival modelbased upon the generalized boarding model using the following equation:f(x;p _((t) _(j) _(−t) _(j-1) ₎,λ_(i)(t _(j) −t _(j-1)))=p _((t) _(j)_(−t) _(j-1) ₎ I(x=0)+(1−p _((t) _(j) _(−t) _(j- 1) ₎)Pois(x,λ _(i)(t_(j) −t _(j-1))), i=1, . . . ,n; j=1, . . . ,T  (3)where T represents the total time interval between scheduled arrivals ateach particular stop.

Thus, the current equations model a Poisson distribution of demand thatvaries by time and the geography of a stop. Additionally, the equationsinclude an assumption that a latent probability of no demand varies overtime to absorb the zero values of no one being at the bus stop at agiven time. Zero values may be both a result of the automatic passengercounter registering a zero count at a stop (i.e., the vehicle onlystopped to allow passengers to depart) as well as a non-count for a stop(i.e., the vehicle did not stop as there were no passengers wishing toboard or depart). In this sense, the lack of demand can be accuratelymodeled to expose true demand where people may or may not use thetransportation system (e.g., catch a bus) at that place and time (e.g.,a particular time during a specific day).

To gain further insight from the data, additional explanatory variablesmay be introduced in the form of random effect terms for both the hourof the day and day of the week. Such an introduction can allow forinsight into how instantaneous demand λ_(i) and the probability of nodemand p_((t) _(j) _(−t) _(j-1) ₎ vary by the hour of the day and theday of the week. This provides for a modeling process where asignificant amount of variation is explained and accounted for.

The statistical model of the demand information may be fitted usingBayesian methods and may incorporate an observation process, i.e., themodel compensates for the possibility that passenger arrival informationfrom the automatic passenger counter may include errors. An example ofthe model may be:N _(j)˜Poisson((1−u _(j))λ_(j)(t _(j) −t _(j-1)))u _(j)˜Bernoulli(p _(j))λ_(j)=exp(a[h(t _(j))]+b[w(t _(j))])p _(j)=1/(1+exp(−c[h(t _(j))]−d[w(t _(j))])))Y _(j)˜Normal((1−u _(j))N _(j),τ)  (4)

In the above model, Y_(j) is the count of passengers at bus stop ibetween times t_(j-1) and t_(j); p_(j) is the probability of no one evenwanting to ride the vehicle between times t_(j-1) and t_(j); h is afunction that returns the hour of a time in integers from 1 to 20 (e.g.,assuming that there are 4 morning hours in which the transportationsystem does not operate); w is a function that returns the weekday of atime in integers from 1 to 7; a and b are 20×7 matrices that capturesthe interactions of hour and weekday, which are estimated from the data(again, limited to 20 in this example as it is assumed thetransportation system operates 20 hours a day); c and d are similar 20×7matrices that captures the interactions of hour and weekday, which areestimated from the data; τ is the precision of the counting system bymeasuring the random variation not explained by the time varying model.

Additionally, as modeled, N_(j) represents a listing of the number ofpassengers without accounting for error and u_(j) represents whether aperson is actually waiting at a stop or not at a given time. Thus, asmodeled, the passenger count Y_(j) for each stop accounts for any errorgenerated by the automatic passenger counters by including both u_(j)and τ.

Additionally, based upon the data provided for and the models includedabove in equation set (4), a computing device can model 108 theinstantaneous demand for a stop λ_(i) as well as model 110 theprobability p_(j) that there is no demand at the stop, i.e., nopassengers are waiting to board at that stop at a specific time. Morespecifically, the computing device can model 108 the instantaneousdemand λ_(i) as a function of the expected rate of passenger arrivalover time for each hour of the day and each day of the week. Similarly,the computing device can model 110 the probability p_(j) of no demand asan inverse function of the expected rate of passenger arrival for eachhour of the day and each day of the week.

In an example, the statistical models may be fit using a Bayesianfitting method such as a Markov Chain Monte Carlo method. Markov ChainMonte Carlo methods represent a class of algorithms for sampling fromprobability distributions based upon constructing a Markov Chain, i.e.,a data structure where a current state is dependent only upon itself,not any previous states of the data. Such a data structure is applicableto modeling demand as the demand for a particular time period may bebased solely upon the time of day and the day of the week for thatperiod, and be may be completely independent of any surrounding periods.More specifically, the demand at a stop in a transportation systembetween 9:00 and 10:00 can be completely independent of the demand atthat stop between 8:00 and 9:00 as well as between 10:00 and 11:00.

The computing device may generate 112 a report including the specificdemand information. The report may include, for example, arepresentation or indication of the instantaneous demand model asdetermined from the instantaneous demand model for a particular stop, arepresentation or indication of the probability that there is no demandas determined based upon the probability of no demand model for theparticular stop, as well as additional information such as estimated andactual passenger arrival information for the stop. The report may bedistributed or otherwise presented to one or more recipients for furtheranalysis and review. For example, a scheduling manager for a publictransportation system may analyze both the instantaneous demand modelfor a stop as well as the probability for no demand model at that stop,and determine whether to eliminate that stop from the route at one ormore times of the day or days of the week. Similarly, if theinstantaneous demand model indicates high demand for a stop, thescheduling manager may increase the number of vehicles stopping at thatstop or provide larger vehicles at that stop to accommodate theincreased demand.

It should be noted that the computing device that performed the variousmodel determinations may generate 112 and provide the report. However,this is shown by way of example, and additional computing devices maygenerate the report. For example, a manager may access a reportingapplication at their personal computing device. The personal computingdevice may access a central server, requesting the various demand modelsas described above. The personal computing device, via the reportingapplication or another similar software application, may then generatethe reports for review by the manager.

In addition to generating a report, the system may be configured toperform additional automated operations related to the overall operationof the transportation system. For example, the system may analyze theoverall demand data to adjust one or more particular routes byeliminating stops, creating new stops, and consolidating stops to smoothout demand over time. The system may automatically revise and update aroute schedule, notifying a dispatcher for the transportation systemthat the route changes have been made to accommodate demand. Similarly,the demand information may be used by the system to automatically adjustfares during periods of varying demand. For example, during low demandperiods the system may automatically lower fares to boost demand.Conversely, during high demand periods, the system may maintain a higherfare to maximize income. Additionally, the demand information may beused to automatically determine what type of vehicle to use duringspecific times of the day. For example, during low demand periods, thesystem may schedule a lower capacity vehicle to handle the lower demand,e.g., a 15-20 seat paratransit vehicle. Conversely, during high demandperiods, the system may schedule a higher capacity vehicle such as alarge, articulate bus to handle higher demand.

Additionally, the information contained within the report may beanalyzed to identify and/or anticipate random high demand periods, andto respond accordingly. For example, the system may automatically deployanother vehicle to handle a random period of high demand by contacting adispatcher or a manager in the transportation system, identifying theparticular area of high demand as well as the instruction to deployanother vehicle.

It should be noted that while buses and similar transportation vehiclesare described above, the system is not limited to transportation systemsthat include buses only. Airlines can utilize the automated demandmonitoring techniques as described herein to modify plane schedules toaccommodate passenger demand. Similarly, transportation systemsincluding trains can use the automated demand monitoring techniques toaccommodate for changes in demand as well, by alternating schedules orchanging the types of trains used during certain periods of the day. Forexample, one or more passenger cars can be added to a train during highdemand to accommodate additional riders and maximize potential income.Conversely, during periods of lower demand, one or more passenger carscan be removed from the trains, thereby reducing operating expensesduring those times.

FIGS. 2, 3A and 3B illustrate various components that may be included inan example report as described above. For example, as shown in FIG. 2,the report may include a plot of observed and estimated passengersboarding as modeled by the boarding count model. This information can beused to determine how accurate the boarding count model is with regardto a particular stop being analyzed.

As shown in FIG. 3A, the representation or indication of theinstantaneous demand for a particular stop can be illustrated as a levelplot having the hour of the day on the x-axis, and the day of the weekon the y-axis. Such a plot as that shown in FIG. 3A can provide a quickvisual indication of the instantaneous demand model, providing ascheduling manager or other related personnel a quick indication ofdemand at the stop.

For example, as shown in FIG. 3A, the instantaneous demand at Tuesdaybetween 6:00 and 7:00 is about 30 people. Conversely, the instantaneousdemand at Sunday between 20:00 and 21:00 is less than 10 people. Basedupon this information, the scheduling manager or other related personnelcan make informed decisions as to whether service to a stop should beincreased or decreased at particular times.

Similarly, as shown in FIG. 3B, the representation or indication of theprobability of no demand for a particular stop can be illustrated as alevel plot with hour of the day on the x-axis, and the day of the weekon the y-axis. Such a plot as that shown in FIG. 3B can provide a quickvisual indication of the probability of there being no demand at a stop,providing a scheduling manager or other related personnel a quickindication of when demand for a stop is at its highest and, conversely,when demand for a stop is at its lowest.

For example, as shown in FIG. 3B, the instantaneous demand at Tuesdayaround 19:00 is about 0.7, indicating there is a high probability ofthere being no demand at the stop. Conversely, the instantaneous demandon Monday between 7:00 and 18:00 is less close to zero, indicating thereis a very low probability of there being no demand during those times.Based upon this information, the scheduling manager or other relatedpersonnel can make informed decisions as to whether a stop can beeliminated at particular times without impacting or inconveniencing alarge number of people.

It should be noted that, as described above, the transportation systemmay only operate a set number of hours a day. In the example illustratedin FIGS. 3A and 3B, the transportation system does not operate between1:00 and 5:00 and, thus, those times are not illustrated in the plots.

The processes as described herein, including the model generations,calculations and derivations as described above, may be performed andimplemented by one or more operators of one or more computing deviceslocated at an operations center (e.g., a central operations center for apublic transportation provider). Alternatively, the processes asdescribed herein may be performed automatically by one or more computingdevices.

FIG. 4 depicts a block diagram of internal hardware that may be used tocontain or implement the various computer processes and systems asdiscussed above. An electrical bus 400 serves as the main informationhighway interconnecting the other illustrated components of thehardware. CPU 405 is the central processing unit of the system,performing calculations and logic operations required to execute aprogram. CPU 405, alone or in conjunction with one or more of the otherelements disclosed in FIG. 4, is a processing device, computing deviceor processor as such terms are used within this disclosure. Read onlymemory (ROM) 410 and random access memory (RAM) 415 constitute examplesof memory devices.

A controller 420 interfaces with one or more optional memory devices 425to the system bus 400. These memory devices 425 may include, forexample, an external or internal DVD drive, a CD ROM drive, a harddrive, flash memory, a USB drive or the like. As indicated previously,these various drives and controllers are optional devices. Additionally,the memory devices 425 may be configured to include individual files forstoring any software modules or instructions, auxiliary data, incidentdata, common files for storing groups of contingency tables and/orregression models, or one or more databases for storing the informationas discussed above.

Program instructions, software or interactive modules for performing anyof the functional steps associated with the processes as described abovemay be stored in the ROM 410 and/or the RAM 415. Optionally, the programinstructions may be stored on a tangible computer readable medium suchas a compact disk, a digital disk, flash memory, a memory card, a USBdrive, an optical disc storage medium, a distributed computer storageplatform such as a cloud-based architecture, and/or other recordingmedium

An optional display interface 430 may permit information from the bus400 to be displayed on the display 435 in audio, visual, graphic oralphanumeric format. Communication with external devices may occur usingvarious communication ports 440. A communication port 440 may beattached to a communications network, such as the Internet or a localarea network.

The hardware may also include an interface 445 which allows for receiptof data from input devices such as a keyboard 450 or other input device455 such as a mouse, a joystick, a touch screen, a remote control, apointing device, a video input device and/or an audio input device.

Various of the above-disclosed and other features and functions, oralternatives thereof, may be combined into many other different systemsor applications. Various presently unforeseen or unanticipatedalternatives, modifications, variations or improvements therein may besubsequently made by those skilled in the art, each of which is alsointended to be encompassed by the disclosed embodiments.

What is claimed is:
 1. A method of identifying demand in atransportation system, the method comprising: by one or more automaticpassenger counters associated with one or more vehicles in atransportation system, determining passenger arrival information for aplurality of passengers arriving at a plurality of stops on a route inthe transportation system; determining, by a processing device, aboarding count model based upon the passenger arrival information;determining, by the processing device, a geographic and time-specificgeneralized boarding model based upon the boarding count model as wellas information related to the plurality of stops on the route in thetransportation system; for each of the plurality of stops: determining,by the processing device, an approximated uniform arrival model basedupon the generalized boarding model and a time period between arrivingvehicles at a specific stop, determining, by the processing device, aninstantaneous demand model as a function of an expected rate ofpassenger arrival over time based upon the uniform arrival model byfitting the uniform arrival model using a Bayesian fitting method,wherein fitting the uniform arrival model comprises compensating forerror associated with the passenger arrival information, determining, bythe processing device, a probability of no demand model as an inversefunction of the expected rate of passenger arrival over time based uponthe uniform arrival model, determining an indication of instantaneousdemand based upon the instantaneous demand model, and determining anindication of probability of no demand based upon the probability of nodemand model; and based on the indication of instantaneous demand andthe indication of probability of no demand, automatically altering atleast one operational parameter of the transportation system byperforming at least one of: eliminating a stop on the route in thetransportation system, adding a stop to the route in the transportationsystem, consolidating stops on the route in the transportation system,adjusting fares collected during a specific time period, adjustingvehicle type being dispatched during a specific time period, ordeploying an additional vehicle.
 2. The method of claim 1, whereindetermining the boarding count model comprises modeling passengerarrival using a zero inflated Poisson distribution.
 3. The method ofclaim 2, wherein determining the generalized boarding model comprisesgeneralizing the zero inflated Poisson distribution across each of theplurality of stops for the route in the transportation system.
 4. Themethod of claim 1, wherein the Bayesian fitting method comprises aMarkov Chain Monte Carlo method.
 5. The method of claim 1, wherein thefitted model comprises a demand model varying over time for each of theplurality of stops.
 6. A system for identifying demand in atransportation system, the system comprising: a processing device;automatic passenger counters associated with one or more vehicles in thetransportation system, the automatic passenger counters are configuredto determine passenger arrival information for a plurality of passengersarriving at a plurality of stops on a route in the transportationsystem; and a computer readable medium in communication with theprocessing device, the computer readable medium comprising one or moreprogramming instructions for causing the processing device to: determinea boarding count model based upon the passenger arrival information,determine a geographic and time-specific generalized boarding modelbased upon the boarding count model as well as information related tothe plurality of stops on the route in the transportation system, foreach of the plurality of stops: determine an approximated uniformarrival model based upon the generalized boarding model and a timeperiod between arriving vehicles at a specific stop, determine aninstantaneous demand model as a function of an expected rate ofpassenger arrival over time based upon the uniform arrival model byfitting the uniform arrival model using a Bayesian method, whereinfitting the uniform arrival model comprises compensating for errorassociated with the passenger arrival information, determine aprobability of no demand model as an inverse function of the expectedrate of passenger arrival over time based upon the uniform arrivalmodel, determine an indication of instantaneous demand based upon theinstantaneous demand model, and determine an indication of probabilityof no demand based upon the probability of no demand model; and based onthe indication of instantaneous demand and the indication of probabilityof no demand, automatically alter at least one operational parameter ofthe transportation system by performing at least one of: eliminating astop on the route in the transportation system, adding a stop to theroute in the transportation system, consolidating stops on the route inthe transportation system, adjusting fares collected during a specifictime period, adjusting vehicle type being dispatched during a specifictime period, or deploying an additional vehicle.
 7. The system of claim6, wherein the one or more instructions for causing the processingdevice to determine the boarding count model comprise one or moreinstructions for causing the processing device to model passengerarrival using a zero inflated Poisson distribution.
 8. The system ofclaim 7, wherein the one or more instructions for causing the processingdevice to determine the generalized boarding model comprise one or moreinstructions for causing the processing device to generalize the zeroinflated Poisson distribution across each of the plurality of stops forthe route in the transportation system.
 9. The system of claim 6,wherein the Bayesian fitting method comprises a Markov Chain Monte Carlomethod.
 10. The system of claim 6, wherein the fitted model comprises ademand model varying over time for each of the plurality of stops. 11.The method of claim 1, wherein compensating for error associated withthe passenger arrival information comprises modeling an instantaneousdemand with a normal distribution having a mean and precision, wherein:the mean is based on at least whether a passenger is actually waiting atthe stop at any giving time; and the precision is based on a measurementof random variations in the passenger arrival information.
 12. Thesystem of claim 6, the one or more instructions for causing theprocessing device to compensate for error associated with the passengerarrival information comprise one or more instructions for causing theprocessing device model an instantaneous demand with a normaldistribution having a mean and precision, wherein: the mean is based onat least whether a passenger is actually waiting at the stop at anygiving time; and the precision is based on a measurement of randomvariations in the passenger arrival information.